The feasibility of complete photoionization experiments, in which the full set of photoionization matrix elements is determined, using multiphoton ionization schemes with polarization-shaped pulses has recently been demonstrated [P. Hockett, Phys. Rev. Lett. 112, 223001 (2014)10.1103/PhysRevLett.112.223001]. Here we extend our previous work to discuss further details of the numerics and analysis methodology utilized and compare the results directly to new tomographic photoelectron measurements, which provide a more sensitive test of the validity of the results. In so doing we discuss in detail the physics of the photoionization process and suggest various avenues and prospects for this coherent multiplexing methodology.